Emerging 3D scene representations, such as Neural Radiance Fields (NeRF) and 3D Gaussian Splatting (3DGS), have demonstrated their effectiveness in Simultaneous Localization and Mapping (SLAM) for photo-realistic rendering, particularly when using high-quality video sequences as input. However, existing methods struggle with motion-blurred frames, which are common in real-world scenarios like low-light or long-exposure conditions. This often results in a significant reduction in both camera localization accuracy and map reconstruction quality. To address this challenge, we propose a dense visual deblur SLAM pipeline (i.e. MBA-SLAM) to handle severe motion-blurred inputs and enhance image deblurring. Our approach integrates an efficient motion blur-aware tracker with either neural radiance fields or Gaussian Splatting based mapper. By accurately modeling the physical image formation process of motion-blurred images, our method simultaneously learns 3D scene representation and estimates the cameras' local trajectory during exposure time, enabling proactive compensation for motion blur caused by camera movement. In our experiments, we demonstrate that MBA-SLAM surpasses previous state-of-the-art methods in both camera localization and map reconstruction, showcasing superior performance across a range of datasets, including synthetic and real datasets featuring sharp images as well as those affected by motion blur, highlighting the versatility and robustness of our approach. Code is available at https://github.com/WU-CVGL/MBA-SLAM.